1. The Rendering Equation
Direct (local) illumination
Light directly from light sources
No shadows
Indirect (global) illumination
Hard and soft shadows
Diffuse interreflections (radiosity)
Glossy interreflections (caustics)

2. Early Radiosity

3. Lighting Effects Hard Shadows Soft Shadows Caustics
Indirect Illumination

4. Challenge To evaluate the reflection equation
the incoming radiance must be known
To evaluate the incoming radiance
the reflected radiance must be known

5. To The Rendering Equation
Questions
1. How is light measured?
2. How is the spatial distribution of light energy described?
3. How is reflection from a surface characterized?
4. What are the conditions for equilibrium flow of light in an environment?

6. The Grand Scheme Light and Radiometry Energy Balance Surface
Rendering Equation
Volume
Rendering Equation
Radiosity Equation

7. [outgoing] - [incoming] = [emitted] - [absorbed]
Balance Equation
Accountability
[outgoing] - [incoming] = [emitted] - [absorbed]
Macro level
The total light energy put into the system must equal the energy leaving the system (usually, via heat).
Micro level
The energy flowing into a small region of phase space must equal the energy flowing out.

8. Surface Balance Equation
[outgoing] = [emitted] + [reflected]

9. Direction Conventions
BRDF
Surface vs. Field Radiance

10. Surface Balance Equation
[outgoing] = [emitted] + [reflected] + [transmitted]
BTDF

11. Two-Point Geometry
Ray Tracing

12. Coupling Equations
Invariance of radiance

13. The Rendering Equation
Directional form
Transport operator
i.e. ray tracing
Integrate over
hemisphere of
directions

14. The Rendering Equation
Surface form
Geometry term
Integrate over
all surfaces
Visibility term

15. The Radiosity Equation
Assume diffuse reflection 1. 2.
Question: How to incorporate directional light sources into this framework

16. Integral Equations Integral equations of the 1st kind
Integral equations of the 2nd kind

17. Linear Operators Linear operators act on functions
like matrices act on vectors
They are linear in that
Types of linear operators

18. Rendering Operators
Scattering operator
Transport operator

19. Solving the Rendering Equation
Solution

20. Formal Solution
Neumann series
Verify

21. Successive Approximations
Converged

22. Successive Approximation

23. Light Path

24. Light Path

25. Light Paths

26. Light Transport Integrate over all paths of all lengths Question:
How to sample space of paths?

27. Classic Ray Tracing
From Heckbert
Forward (from eye): E S* (D|G) L

28. Photon Paths
Radiosity
Caustics
From Heckbert

29. How to Solve It? Finite element methods Classic radiosity
Mesh surfaces
Piecewise constant basis functions
Solve matrix equation
Not practical for rendering equation
Monte Carlo methods
Path tracing (distributed ray tracing)
Randomly trace ray from the eye
Bidirectional ray tracing
Photon mapping